R Notebook

###Read the data and split into train and test set

ucars<-read.csv("~/Downloads/usedcars.csv")
head(ucars)

set.seed(30005)

trainRec<-sort(sample(1:nrow(ucars),size=72))

testRec<-setdiff(1:nrow(ucars),trainRec)

traincars<-ucars[trainRec,]

testcars<-ucars[testRec,]

#Introduce a variable shows used cars' age
traincars$Age <- 2025 - traincars$Year
testcars$Age <- 2025 - testcars$Year

#For readability, Introduce a variable to show price in 100,000s
traincars$price_in_100000 <- traincars$Price/100000
testcars$price_in_100000 <- testcars$Price/100000

#Find the relationship between price and numeric variables:Kilometers_Driven, Mileage, Engine, Power

par(mfrow = c(2, 2))

numeric_vars = c('Kilometers_Driven','Mileage','Engine','Power')

# Base R version (one-by-one)
for (var in numeric_vars) {
  plot(traincars[[var]], traincars$price_in_100000,
       xlab = var, ylab = "Price in 100,000s",
       main = paste("Price vs", var),
       pch = 19, col = "steelblue")
}

###Find the relationship between price and categorical variables using boxplot

par(mfrow = c(3, 1))  

boxplot(price_in_100000 ~ Fuel_Type, data = traincars, 
        main = "Price by Fuel Type", col = "lightblue")

boxplot(price_in_100000 ~ Transmission, data = traincars, 
        main = "Price by Transmission", col = "lightgreen")

boxplot(price_in_100000 ~ Owner_Type, data = traincars, 
        main = "Price by Owner Type", col = "lightpink")

###Find the relationship between price and categorical variables using boxplot

par(mfrow = c(3, 1)) 

boxplot(price_in_100000 ~ Brand, data = traincars, 
        main = "Price by Brand", col = "lightgoldenrod")

boxplot(price_in_100000 ~ Year, data = traincars, 
        main = "Price by Year", col = "palegreen")

boxplot(price_in_100000 ~ Seats, data = traincars, 
        main = "Price by number of seats", col = "yellow")

###(3) ###transform categorical variables Agem Fuel_Type, Owner_Type, Transmission to numerical variables

traincars$Owner_Type <- factor(traincars$Owner_Type, levels = c("First", "Second", "Third"), labels = c(1, 2, 3))

###choosing predictors using AIC step-wise method:

nullModel<-lm(Price~1, data = traincars) #Model only with intercept

fullModel <- lm(Price ~ Brand + Age + Kilometers_Driven + Fuel_Type + Transmission + Owner_Type + Mileage + Engine + Power + Seats, data=traincars)

step(nullModel, scope=list(lower=nullModel, upper=fullModel),direction="both")

Start:  AIC=1990.5
Price ~ 1

                    Df  Sum of Sq        RSS    AIC
+ Power              1 5.1251e+13 1.9823e+13 1900.6
+ Brand             10 5.2312e+13 1.8762e+13 1914.6
+ Engine             1 3.5792e+13 3.5282e+13 1942.1
+ Transmission       1 3.5181e+13 3.5892e+13 1943.3
+ Mileage            1 2.7868e+13 4.3206e+13 1956.7
+ Owner_Type         2 1.0999e+13 6.0075e+13 1982.4
+ Age                1 5.5270e+12 6.5547e+13 1986.7
<none>                            7.1074e+13 1990.5
+ Fuel_Type          1 6.2500e+11 7.0449e+13 1991.9
+ Seats              1 1.3638e+10 7.1060e+13 1992.5
+ Kilometers_Driven  1 6.0844e+09 7.1068e+13 1992.5

Step:  AIC=1900.57
Price ~ Power

                    Df  Sum of Sq        RSS    AIC
+ Brand             10 1.4829e+13 4.9940e+12 1821.3
+ Transmission       1 4.6105e+12 1.5213e+13 1883.5
+ Fuel_Type          1 1.7123e+12 1.8111e+13 1896.1
+ Seats              1 1.2818e+12 1.8541e+13 1897.8
<none>                            1.9823e+13 1900.6
+ Owner_Type         2 9.7582e+11 1.8847e+13 1900.9
+ Mileage            1 3.9791e+11 1.9425e+13 1901.1
+ Age                1 1.4236e+11 1.9681e+13 1902.0
+ Engine             1 9.9061e+10 1.9724e+13 1902.2
+ Kilometers_Driven  1 2.7399e+09 1.9820e+13 1902.6
- Power              1 5.1251e+13 7.1074e+13 1990.5

Step:  AIC=1821.31
Price ~ Power + Brand

                    Df  Sum of Sq        RSS    AIC
+ Fuel_Type          1 9.4813e+11 4.0459e+12 1808.2
+ Seats              1 6.1340e+11 4.3806e+12 1813.9
+ Transmission       1 4.6484e+11 4.5292e+12 1816.3
+ Mileage            1 2.4633e+11 4.7477e+12 1819.7
+ Engine             1 1.5615e+11 4.8379e+12 1821.0
<none>                            4.9940e+12 1821.3
+ Kilometers_Driven  1 7.5671e+10 4.9183e+12 1822.2
+ Age                1 3.0485e+10 4.9635e+12 1822.9
+ Owner_Type         2 1.1992e+11 4.8741e+12 1823.6
- Brand             10 1.4829e+13 1.9823e+13 1900.6
- Power              1 1.3768e+13 1.8762e+13 1914.6

Step:  AIC=1808.15
Price ~ Power + Brand + Fuel_Type

                    Df  Sum of Sq        RSS    AIC
+ Mileage            1 2.6867e+11 3.7772e+12 1805.2
+ Transmission       1 2.3113e+11 3.8148e+12 1805.9
+ Seats              1 1.9533e+11 3.8505e+12 1806.6
<none>                            4.0459e+12 1808.2
+ Age                1 3.4376e+10 4.0115e+12 1809.5
+ Kilometers_Driven  1 1.5652e+10 4.0302e+12 1809.9
+ Engine             1 1.2654e+09 4.0446e+12 1810.1
+ Owner_Type         2 3.6448e+10 4.0094e+12 1811.5
- Fuel_Type          1 9.4813e+11 4.9940e+12 1821.3
- Brand             10 1.4065e+13 1.8111e+13 1896.1
- Power              1 1.4380e+13 1.8426e+13 1915.3

Step:  AIC=1805.2
Price ~ Power + Brand + Fuel_Type + Mileage

                    Df  Sum of Sq        RSS    AIC
+ Transmission       1 3.2901e+11 3.4482e+12 1800.6
+ Kilometers_Driven  1 1.1473e+11 3.6625e+12 1805.0
<none>                            3.7772e+12 1805.2
+ Age                1 9.9348e+10 3.6779e+12 1805.3
+ Seats              1 5.4295e+10 3.7229e+12 1806.2
+ Engine             1 4.1425e+10 3.7358e+12 1806.4
- Mileage            1 2.6867e+11 4.0459e+12 1808.2
+ Owner_Type         2 1.4600e+10 3.7626e+12 1808.9
- Fuel_Type          1 9.7046e+11 4.7477e+12 1819.7
- Power              1 7.9986e+12 1.1776e+13 1885.1
- Brand             10 1.3729e+13 1.7507e+13 1895.6

Step:  AIC=1800.64
Price ~ Power + Brand + Fuel_Type + Mileage + Transmission

                    Df  Sum of Sq        RSS    AIC
+ Kilometers_Driven  1 1.0769e+11 3.3405e+12 1800.3
+ Seats              1 1.0459e+11 3.3436e+12 1800.4
<none>                            3.4482e+12 1800.6
+ Age                1 5.9766e+10 3.3884e+12 1801.4
+ Engine             1 2.3657e+10 3.4245e+12 1802.1
+ Owner_Type         2 3.2029e+10 3.4162e+12 1804.0
- Transmission       1 3.2901e+11 3.7772e+12 1805.2
- Mileage            1 3.6655e+11 3.8148e+12 1805.9
- Fuel_Type          1 6.9992e+11 4.1481e+12 1811.9
- Power              1 4.5691e+12 8.0173e+12 1859.4
- Brand             10 9.9217e+12 1.3370e+13 1878.2

Step:  AIC=1800.35
Price ~ Power + Brand + Fuel_Type + Mileage + Transmission + 
    Kilometers_Driven

                    Df  Sum of Sq        RSS    AIC
<none>                            3.3405e+12 1800.3
+ Seats              1 8.0771e+10 3.2597e+12 1800.6
- Kilometers_Driven  1 1.0769e+11 3.4482e+12 1800.6
+ Engine             1 2.9204e+10 3.3113e+12 1801.7
+ Age                1 1.9443e+09 3.3386e+12 1802.3
+ Owner_Type         2 6.0638e+10 3.2799e+12 1803.0
- Transmission       1 3.2197e+11 3.6625e+12 1805.0
- Mileage            1 4.6644e+11 3.8070e+12 1807.8
- Fuel_Type          1 8.0407e+11 4.1446e+12 1813.9
- Power              1 4.4575e+12 7.7980e+12 1859.4
- Brand             10 9.2120e+12 1.2552e+13 1875.7

Call:
lm(formula = Price ~ Power + Brand + Fuel_Type + Mileage + Transmission + 
    Kilometers_Driven, data = traincars)

Coefficients:
       (Intercept)               Power            BrandBMW           BrandFord          BrandHonda        BrandHyundai       BrandMahindra         BrandMaruti  
         2.477e+06           5.657e+03           4.238e+04          -9.910e+05          -8.947e+05          -1.166e+06          -1.194e+06          -9.918e+05  
     BrandMercedes           BrandTata         BrandToyota     BrandVolkswagen     Fuel_TypePetrol             Mileage  TransmissionManual   Kilometers_Driven  
         2.926e+03          -8.977e+05          -7.333e+05          -8.508e+05          -2.632e+05          -4.236e+04          -2.207e+05          -6.367e+00  

###choosing predictors using BIC step-wise method:

step(nullModel, scope=list(lower=nullModel, upper=fullModel),direction="both",k=log(nrow(traincars)))

Start:  AIC=1992.78
Price ~ 1

                    Df  Sum of Sq        RSS    AIC
+ Power              1 5.1251e+13 1.9823e+13 1905.1
+ Brand             10 5.2312e+13 1.8762e+13 1939.7
+ Engine             1 3.5792e+13 3.5282e+13 1946.6
+ Transmission       1 3.5181e+13 3.5892e+13 1947.9
+ Mileage            1 2.7868e+13 4.3206e+13 1961.2
+ Owner_Type         2 1.0999e+13 6.0075e+13 1989.2
+ Age                1 5.5270e+12 6.5547e+13 1991.2
<none>                            7.1074e+13 1992.8
+ Fuel_Type          1 6.2500e+11 7.0449e+13 1996.4
+ Seats              1 1.3638e+10 7.1060e+13 1997.0
+ Kilometers_Driven  1 6.0844e+09 7.1068e+13 1997.0

Step:  AIC=1905.12
Price ~ Power

                    Df  Sum of Sq        RSS    AIC
+ Brand             10 1.4829e+13 4.9940e+12 1848.6
+ Transmission       1 4.6105e+12 1.5213e+13 1890.3
+ Fuel_Type          1 1.7123e+12 1.8111e+13 1902.9
+ Seats              1 1.2818e+12 1.8541e+13 1904.6
<none>                            1.9823e+13 1905.1
+ Mileage            1 3.9791e+11 1.9425e+13 1907.9
+ Age                1 1.4236e+11 1.9681e+13 1908.9
+ Engine             1 9.9061e+10 1.9724e+13 1909.0
+ Kilometers_Driven  1 2.7399e+09 1.9820e+13 1909.4
+ Owner_Type         2 9.7582e+11 1.8847e+13 1910.0
- Power              1 5.1251e+13 7.1074e+13 1992.8

Step:  AIC=1848.63
Price ~ Power + Brand

                    Df  Sum of Sq        RSS    AIC
+ Fuel_Type          1 9.4813e+11 4.0459e+12 1837.7
+ Seats              1 6.1340e+11 4.3806e+12 1843.5
+ Transmission       1 4.6484e+11 4.5292e+12 1845.9
<none>                            4.9940e+12 1848.6
+ Mileage            1 2.4633e+11 4.7477e+12 1849.3
+ Engine             1 1.5615e+11 4.8379e+12 1850.6
+ Kilometers_Driven  1 7.5671e+10 4.9183e+12 1851.8
+ Age                1 3.0485e+10 4.9635e+12 1852.5
+ Owner_Type         2 1.1992e+11 4.8741e+12 1855.4
- Brand             10 1.4829e+13 1.9823e+13 1905.1
- Power              1 1.3768e+13 1.8762e+13 1939.7

Step:  AIC=1837.74
Price ~ Power + Brand + Fuel_Type

                    Df  Sum of Sq        RSS    AIC
+ Mileage            1 2.6867e+11 3.7772e+12 1837.1
<none>                            4.0459e+12 1837.7
+ Transmission       1 2.3113e+11 3.8148e+12 1837.8
+ Seats              1 1.9533e+11 3.8505e+12 1838.5
+ Age                1 3.4376e+10 4.0115e+12 1841.4
+ Kilometers_Driven  1 1.5652e+10 4.0302e+12 1841.7
+ Engine             1 1.2654e+09 4.0446e+12 1842.0
+ Owner_Type         2 3.6448e+10 4.0094e+12 1845.7
- Fuel_Type          1 9.4813e+11 4.9940e+12 1848.6
- Brand             10 1.4065e+13 1.8111e+13 1902.9
- Power              1 1.4380e+13 1.8426e+13 1942.6

Step:  AIC=1837.07
Price ~ Power + Brand + Fuel_Type + Mileage

                    Df  Sum of Sq        RSS    AIC
+ Transmission       1 3.2901e+11 3.4482e+12 1834.8
<none>                            3.7772e+12 1837.1
- Mileage            1 2.6867e+11 4.0459e+12 1837.7
+ Kilometers_Driven  1 1.1473e+11 3.6625e+12 1839.1
+ Age                1 9.9348e+10 3.6779e+12 1839.4
+ Seats              1 5.4295e+10 3.7229e+12 1840.3
+ Engine             1 4.1425e+10 3.7358e+12 1840.6
+ Owner_Type         2 1.4600e+10 3.7626e+12 1845.3
- Fuel_Type          1 9.7046e+11 4.7477e+12 1849.3
- Brand             10 1.3729e+13 1.7507e+13 1904.7
- Power              1 7.9986e+12 1.1776e+13 1914.7

Step:  AIC=1834.79
Price ~ Power + Brand + Fuel_Type + Mileage + Transmission

                    Df  Sum of Sq        RSS    AIC
<none>                            3.4482e+12 1834.8
+ Kilometers_Driven  1 1.0769e+11 3.3405e+12 1836.8
+ Seats              1 1.0459e+11 3.3436e+12 1836.8
- Transmission       1 3.2901e+11 3.7772e+12 1837.1
- Mileage            1 3.6655e+11 3.8148e+12 1837.8
+ Age                1 5.9766e+10 3.3884e+12 1837.8
+ Engine             1 2.3657e+10 3.4245e+12 1838.6
+ Owner_Type         2 3.2029e+10 3.4162e+12 1842.7
- Fuel_Type          1 6.9992e+11 4.1481e+12 1843.8
- Brand             10 9.9217e+12 1.3370e+13 1889.6
- Power              1 4.5691e+12 8.0173e+12 1891.3

Call:
lm(formula = Price ~ Power + Brand + Fuel_Type + Mileage + Transmission, 
    data = traincars)

Coefficients:
       (Intercept)               Power            BrandBMW           BrandFord          BrandHonda        BrandHyundai       BrandMahindra         BrandMaruti  
           2134420                5715               83466             -969348             -935963            -1204552            -1167995             -965189  
     BrandMercedes           BrandTata         BrandToyota     BrandVolkswagen     Fuel_TypePetrol             Mileage  TransmissionManual  
             33754             -901201             -755428             -845437             -222150              -35033             -223043  

###choosing predictors using subset method:

library("leaps")
Bestfits <- regsubsets(Price~ Brand + Age + Kilometers_Driven + Fuel_Type + Transmission + Owner_Type + Mileage + Engine + Power + Seats, data=traincars, nbest=1)
summary(Bestfits)

Subset selection object
Call: regsubsets.formula(Price ~ Brand + Age + Kilometers_Driven + 
    Fuel_Type + Transmission + Owner_Type + Mileage + Engine + 
    Power + Seats, data = traincars, nbest = 1)
20 Variables  (and intercept)
                   Forced in Forced out
BrandBMW               FALSE      FALSE
BrandFord              FALSE      FALSE
BrandHonda             FALSE      FALSE
BrandHyundai           FALSE      FALSE
BrandMahindra          FALSE      FALSE
BrandMaruti            FALSE      FALSE
BrandMercedes          FALSE      FALSE
BrandTata              FALSE      FALSE
BrandToyota            FALSE      FALSE
BrandVolkswagen        FALSE      FALSE
Age                    FALSE      FALSE
Kilometers_Driven      FALSE      FALSE
Fuel_TypePetrol        FALSE      FALSE
TransmissionManual     FALSE      FALSE
Owner_Type2            FALSE      FALSE
Owner_Type3            FALSE      FALSE
Mileage                FALSE      FALSE
Engine                 FALSE      FALSE
Power                  FALSE      FALSE
Seats                  FALSE      FALSE
1 subsets of each size up to 8
Selection Algorithm: exhaustive
         BrandBMW BrandFord BrandHonda BrandHyundai BrandMahindra BrandMaruti BrandMercedes BrandTata BrandToyota BrandVolkswagen Age Kilometers_Driven
1  ( 1 ) " "      " "       " "        " "          " "           " "         " "           " "       " "         " "             " " " "              
2  ( 1 ) " "      " "       " "        " "          " "           " "         " "           " "       " "         " "             " " " "              
3  ( 1 ) " "      " "       " "        "*"          " "           " "         " "           " "       " "         " "             " " " "              
4  ( 1 ) "*"      " "       " "        " "          " "           " "         "*"           " "       " "         " "             " " " "              
5  ( 1 ) "*"      " "       " "        " "          " "           " "         "*"           " "       " "         " "             " " " "              
6  ( 1 ) "*"      " "       " "        "*"          " "           " "         "*"           " "       " "         " "             " " " "              
7  ( 1 ) "*"      " "       " "        "*"          " "           " "         "*"           " "       " "         " "             "*" " "              
8  ( 1 ) " "      "*"       "*"        "*"          "*"           "*"         " "           "*"       " "         "*"             " " " "              
         Fuel_TypePetrol TransmissionManual Owner_Type2 Owner_Type3 Mileage Engine Power Seats
1  ( 1 ) " "             " "                " "         " "         " "     " "    "*"   " "  
2  ( 1 ) " "             "*"                " "         " "         " "     " "    "*"   " "  
3  ( 1 ) " "             "*"                " "         " "         " "     " "    "*"   " "  
4  ( 1 ) " "             "*"                " "         " "         " "     " "    "*"   " "  
5  ( 1 ) " "             "*"                " "         " "         " "     " "    "*"   "*"  
6  ( 1 ) " "             "*"                " "         " "         " "     " "    "*"   "*"  
7  ( 1 ) " "             "*"                " "         " "         " "     " "    "*"   "*"  
8  ( 1 ) " "             " "                " "         " "         " "     " "    "*"   " "  

plot(Bestfits, scale = "r2") #use r2

plot(Bestfits, scale = "adjr2") #Use adj_r2

plot(Bestfits, scale = "bic") #Use bic

#AIC model

model1 <- lm(Price ~  Power + Brand + Fuel_Type + Mileage + Transmission + 
    Kilometers_Driven, data = traincars)
model1

Call:
lm(formula = Price ~ Power + Brand + Fuel_Type + Mileage + Transmission + 
    Kilometers_Driven, data = traincars)

Coefficients:
       (Intercept)               Power            BrandBMW           BrandFord          BrandHonda        BrandHyundai       BrandMahindra         BrandMaruti  
         2.477e+06           5.657e+03           4.238e+04          -9.910e+05          -8.947e+05          -1.166e+06          -1.194e+06          -9.918e+05  
     BrandMercedes           BrandTata         BrandToyota     BrandVolkswagen     Fuel_TypePetrol             Mileage  TransmissionManual   Kilometers_Driven  
         2.926e+03          -8.977e+05          -7.333e+05          -8.508e+05          -2.632e+05          -4.236e+04          -2.207e+05          -6.367e+00  

par(mfrow=c(2,2))
AIC_lm = lm(formula = Price ~ Brand + Fuel_Type + Mileage + Transmission + Kilometers_Driven, data = traincars)
plot(AIC_lm)

#BIC mdoel
model2 <- lm(Price ~  Power + Brand + Fuel_Type + Mileage + Transmission, data = traincars)
model2

Call:
lm(formula = Price ~ Power + Brand + Fuel_Type + Mileage + Transmission, 
    data = traincars)

Coefficients:
       (Intercept)               Power            BrandBMW           BrandFord          BrandHonda        BrandHyundai       BrandMahindra         BrandMaruti  
           2134420                5715               83466             -969348             -935963            -1204552            -1167995             -965189  
     BrandMercedes           BrandTata         BrandToyota     BrandVolkswagen     Fuel_TypePetrol             Mileage  TransmissionManual  
             33754             -901201             -755428             -845437             -222150              -35033             -223043  

par(mfrow=c(2,2))
BIC_lm = lm(formula = Price ~ Brand + Fuel_Type + Mileage + Transmission + Power, data = traincars)
plot(BIC_lm)

#subset model
model3 <- lm(Price ~ Brand + Transmission + Power + Seats, data = traincars)
model3

Call:
lm(formula = Price ~ Brand + Transmission + Power + Seats, data = traincars)

Coefficients:
       (Intercept)            BrandBMW           BrandFord          BrandHonda        BrandHyundai       BrandMahindra         BrandMaruti       BrandMercedes  
            475735               87185             -877747            -1033760            -1175541            -1022604            -1020954               95158  
         BrandTata         BrandToyota     BrandVolkswagen  TransmissionManual               Power               Seats  
           -935651             -764668             -844961             -288212                6374              160413  

par(mfrow=c(2,2))
Sub_lm = lm(formula = Price ~ Brand + Transmission + Power + Seats, data = traincars)
plot(Sub_lm)

#AIC model with only logged power
model1_1 <- lm(Price ~ log(Power) + Brand + Fuel_Type + Mileage + Transmission + 
    Kilometers_Driven, data = traincars)
model1_1

Call:
lm(formula = Price ~ log(Power) + Brand + Fuel_Type + Mileage + 
    Transmission + Kilometers_Driven, data = traincars)

Coefficients:
       (Intercept)          log(Power)            BrandBMW           BrandFord          BrandHonda        BrandHyundai       BrandMahindra         BrandMaruti  
        -2.749e+06           1.209e+06           2.706e+04          -8.947e+05          -7.294e+05          -1.016e+06          -1.110e+06          -8.591e+05  
     BrandMercedes           BrandTata         BrandToyota     BrandVolkswagen     Fuel_TypePetrol             Mileage  TransmissionManual   Kilometers_Driven  
         2.318e+04          -7.739e+05          -6.981e+05          -7.343e+05          -1.802e+05          -3.697e+04          -1.472e+05          -1.118e+01  

#BIC model with only logged power

model2_1 <- lm(Price ~ log(Power) + Brand + Fuel_Type + Mileage + Transmission, data = traincars)
model2_1

Call:
lm(formula = Price ~ log(Power) + Brand + Fuel_Type + Mileage + 
    Transmission, data = traincars)

Coefficients:
       (Intercept)          log(Power)            BrandBMW           BrandFord          BrandHonda        BrandHyundai       BrandMahindra         BrandMaruti  
          -3199388             1191453              108769             -852837             -796458            -1085552            -1057885             -817465  
     BrandMercedes           BrandTata         BrandToyota     BrandVolkswagen     Fuel_TypePetrol             Mileage  TransmissionManual  
             80254             -780618             -739189             -734490             -106670              -26502             -166047  

#Subset model with only logged power

model3_1 <- lm(Price ~ Brand + Transmission + log(Power) + Seats, data = traincars)
model3_1

Call:
lm(formula = Price ~ Brand + Transmission + log(Power) + Seats, 
    data = traincars)

Coefficients:
       (Intercept)            BrandBMW           BrandFord          BrandHonda        BrandHyundai       BrandMahindra         BrandMaruti       BrandMercedes  
          -4976783              103876             -779089             -870159            -1054686             -989824             -821630              126517  
         BrandTata         BrandToyota     BrandVolkswagen  TransmissionManual          log(Power)               Seats  
           -783899             -748247             -697566             -174881             1331982              104265  

#AIC model with both logged price and logged power

model1_2 <- lm(log(Price) ~  log(Power) + Brand + Fuel_Type + Mileage + Transmission + 
    Kilometers_Driven, data = traincars)
model1_2

Call:
lm(formula = log(Price) ~ log(Power) + Brand + Fuel_Type + Mileage + 
    Transmission + Kilometers_Driven, data = traincars)

Coefficients:
       (Intercept)          log(Power)            BrandBMW           BrandFord          BrandHonda        BrandHyundai       BrandMahindra         BrandMaruti  
         1.164e+01           7.248e-01           8.534e-04          -4.748e-01          -3.892e-01          -6.584e-01          -6.853e-01          -4.646e-01  
     BrandMercedes           BrandTata         BrandToyota     BrandVolkswagen     Fuel_TypePetrol             Mileage  TransmissionManual   Kilometers_Driven  
        -3.923e-02          -4.384e-01          -3.386e-01          -3.862e-01          -2.197e-01          -3.302e-02          -2.012e-01          -2.882e-06  

#BIC model with both logged price and logged power

model2_2 <- lm(log(Price) ~  log(Power) + Brand + Fuel_Type + Mileage + Transmission, data = traincars)
model2_2

Call:
lm(formula = log(Price) ~ log(Power) + Brand + Fuel_Type + Mileage + 
    Transmission, data = traincars)

Coefficients:
       (Intercept)          log(Power)            BrandBMW           BrandFord          BrandHonda        BrandHyundai       BrandMahindra         BrandMaruti  
          11.52560             0.72024             0.02191            -0.46404            -0.40652            -0.67650            -0.67184            -0.45387  
     BrandMercedes           BrandTata         BrandToyota     BrandVolkswagen     Fuel_TypePetrol             Mileage  TransmissionManual  
          -0.02452            -0.44012            -0.34923            -0.38629            -0.20070            -0.03032            -0.20603  

#Subset model with both logged price and logged power

model3_2 <- lm(log(Price) ~ Brand + Transmission + log(Power) + Seats, data = traincars)
model3_2

Call:
lm(formula = log(Price) ~ Brand + Transmission + log(Power) + 
    Seats, data = traincars)

Coefficients:
       (Intercept)            BrandBMW           BrandFord          BrandHonda        BrandHyundai       BrandMahindra         BrandMaruti       BrandMercedes  
          9.533864            0.008435           -0.399509           -0.460418           -0.641015           -0.552593           -0.479700            0.014670  
         BrandTata         BrandToyota     BrandVolkswagen  TransmissionManual          log(Power)               Seats  
         -0.458189           -0.341865           -0.371543           -0.229344            0.876573            0.109526  

#Checking for the residual using AIC
predicted_prices <- predict(model1, newdata = testcars)
plot(testcars$Price, predicted_prices,
     xlab = "Actual Price",
     ylab = "Predicted Price",
     main = "Actual vs Predicted Prices",
     pch = 19, col = "blue")
abline(0, 1, col = "red", lwd = 2)  # perfect prediction line

residuals <- testcars$Price - predict(model1, newdata = testcars)

# Plot histogram to see if it is more normal
hist(residuals,
     main = "Histogram of AIC (Actual - Predicted Price)",
     xlab = "Residual (Price - Predicted)",
     col = "skyblue",
     border = "white",
     breaks = 10)

#Checking for the residual using AIC logged power

predicted_prices <- predict(model1_1, newdata = testcars)
plot(testcars$Price, predicted_prices,
     xlab = "Actual Price",
     ylab = "Predicted Price",
     main = "Actual vs Predicted Prices",
     pch = 19, col = "blue")
abline(0, 1, col = "red", lwd = 2)  # perfect prediction line

residuals <- testcars$Price - predict(model1_1, newdata = testcars)

# Plot histogram
hist(residuals,
     main = "Histogram of AIC log power (Actual - Predicted Price)",
     xlab = "Residual (Price - Predicted)",
     col = "skyblue",
     border = "white",
     breaks = 10)

#Checking for the residual using AIC both logged price and logged power

predicted_prices <- predict(model1_2, newdata = testcars)
plot(testcars$Price, predicted_prices,
     xlab = "Actual Price",
     ylab = "Predicted Price",
     main = "Actual vs Predicted Prices",
     pch = 19, col = "blue")
abline(0, 1, col = "red", lwd = 2)  # perfect prediction line

residuals <- testcars$Price - predict(model1_2, newdata = testcars)

# Plot histogram
hist(residuals,
     main = "Histogram of AIC log price and log power",
     xlab = "Residual (Price - Predicted)",
     col = "skyblue",
     border = "white",
     breaks = 10)

#Checking for the residual using BIC

predicted_prices <- predict(model2, newdata = testcars)
plot(testcars$Price, predicted_prices,
     xlab = "Actual Price",
     ylab = "Predicted Price",
     main = "Actual vs Predicted Prices",
     pch = 19, col = "blue")
abline(0, 1, col = "red", lwd = 2)  # perfect prediction line

residuals <- testcars$Price - predict(model2, newdata = testcars)

# Plot histogram
hist(residuals,
     main = "Histogram of BIC (Actual - Predicted Price)",
     xlab = "Residual (Price - Predicted)",
     col = "skyblue",
     border = "white",
     breaks = 10)

#Checking for the residual using BIC power logged

predicted_prices <- predict(model2_1, newdata = testcars)
plot(testcars$Price, predicted_prices,
     xlab = "Actual Price",
     ylab = "Predicted Price",
     main = "Actual vs Predicted Prices",
     pch = 19, col = "blue")
abline(0, 1, col = "red", lwd = 2)  # perfect prediction line

residuals <- testcars$Price - predict(model2_1, newdata = testcars)

# Plot histogram
hist(residuals,
     main = "Histogram of BIC log power (Actual - Predicted Price)",
     xlab = "Residual (Price - Predicted)",
     col = "skyblue",
     border = "white",
     breaks = 10)

#Checking for the residual using BIC both logged price and logged power

predicted_prices <- predict(model2_2, newdata = testcars)
plot(testcars$Price, predicted_prices,
     xlab = "Actual Price",
     ylab = "Predicted Price",
     main = "Actual vs Predicted Prices",
     pch = 19, col = "blue")
abline(0, 1, col = "red", lwd = 2)  # perfect prediction line

residuals <- testcars$Price - predict(model2_2, newdata = testcars)

# Plot histogram
hist(residuals,
     main = "Histogram of BIC log price and log power",
     xlab = "Residual (Price - Predicted)",
     col = "skyblue",
     border = "white",
     breaks = 10)

#Checking for the residual using subset
predicted_prices <- predict(model3, newdata = testcars)
plot(testcars$Price, predicted_prices,
     xlab = "Actual Price",
     ylab = "Predicted Price",
     main = "Actual vs Predicted Prices",
     pch = 19, col = "blue")
abline(0, 1, col = "red", lwd = 2)  # perfect prediction line

residuals <- testcars$Price - predict(model3, newdata = testcars)

# Plot histogram
hist(residuals,
     main = "Histogram of Subset (Actual - Predicted Price)",
     xlab = "Residual (Price - Predicted)",
     col = "skyblue",
     border = "white",
     breaks = 10)

NA
NA

#Checking for the residual using subset power logged

predicted_prices <- predict(model3_1, newdata = testcars)
plot(testcars$Price, predicted_prices,
     xlab = "Actual Price",
     ylab = "Predicted Price",
     main = "Actual vs Predicted Prices",
     pch = 19, col = "blue")
abline(0, 1, col = "red", lwd = 2)  # perfect prediction line

residuals <- testcars$Price - predict(model3_1, newdata = testcars)

# Plot histogram
hist(residuals,
     main = "Histogram of subset log power(Actual - Predicted Price)",
     xlab = "Residual (Price - Predicted)",
     col = "skyblue",
     border = "white",
     breaks = 10)

#Checking for the residual using subset both logged price and logged power

predicted_prices <- predict(model3_2, newdata = testcars)
plot(testcars$Price, predicted_prices,
     xlab = "Actual Price",
     ylab = "Predicted Price",
     main = "Actual vs Predicted Prices",
     pch = 19, col = "blue")
abline(0, 1, col = "red", lwd = 2)  # perfect prediction line

residuals <- testcars$Price - predict(model3_2, newdata = testcars)

# Plot histogram
hist(residuals,
     main = "Histogram of subset log price and log pwoer",
     xlab = "Residual (Price - Predicted)",
     col = "skyblue",
     border = "white",
     breaks = 10)

NA
NA
NA

#Final decision for the model: AIC logged Power and check for the validity

par(mfrow=c(2,2))
loggedAIC_lm = lm(formula = Price ~ log(Power) + Brand + Fuel_Type + Mileage + Transmission + 
    Kilometers_Driven, data = traincars)
plot(loggedAIC_lm)

par(mfrow=c(2,2))
loggedBIC_lm = lm(formula = Price ~ log(Power) + Brand + Fuel_Type + Mileage + Transmission, data = traincars)
plot(loggedBIC_lm)

par(mfrow=c(2,2))
loggedSub_lm = lm(formula = Price ~ Brand + Transmission + log(Power) + Seats, data = traincars)
plot(loggedSub_lm)

###additional information about multicolinearity

#Checking for multicolinearity

library(car)
vif(model1)

                      GVIF Df GVIF^(1/(2*Df))
Power             3.484163  1        1.866591
Brand             6.596049 10        1.098915
Fuel_Type         1.531354  1        1.237479
Mileage           3.084286  1        1.756214
Transmission      2.670094  1        1.634042
Kilometers_Driven 1.834917  1        1.354591

vif(model2)

                 GVIF Df GVIF^(1/(2*Df))
Power        3.469040  1        1.862536
Brand        4.861174 10        1.082274
Fuel_Type    1.253487  1        1.119592
Mileage      2.684819  1        1.638542
Transmission 2.669180  1        1.633762

vif(model3)

                 GVIF Df GVIF^(1/(2*Df))
Brand        3.664279 10        1.067086
Transmission 2.499584  1        1.581007
Power        2.276120  1        1.508682
Seats        1.364545  1        1.168137

###Considering for the cost with all the reasonable predictors

lss_lm <- lm(Price ~ Brand + Fuel_Type + Mileage + Transmission + Power + Kilometers_Driven + Seats, data = traincars)
lss_lm

Call:
lm(formula = Price ~ Brand + Fuel_Type + Mileage + Transmission + 
    Power + Kilometers_Driven + Seats, data = traincars)

Coefficients:
       (Intercept)            BrandBMW           BrandFord          BrandHonda        BrandHyundai       BrandMahindra         BrandMaruti       BrandMercedes  
         1.901e+06           5.948e+04          -9.473e+05          -9.289e+05          -1.147e+06          -1.145e+06          -9.899e+05           3.322e+04  
         BrandTata         BrandToyota     BrandVolkswagen     Fuel_TypePetrol             Mileage  TransmissionManual               Power   Kilometers_Driven  
        -8.923e+05          -7.572e+05          -8.296e+05          -2.201e+05          -3.359e+04          -2.380e+05           5.854e+03          -5.664e+00  
             Seats  
         6.552e+04  

predicted_prices <- predict(lss_lm, newdata = testcars)
plot(testcars$Price, predicted_prices,
     xlab = "Actual Price",
     ylab = "Predicted Price",
     main = "Actual vs Predicted Prices",
     pch = 19, col = "blue")
abline(0, 1, col = "red", lwd = 2)  # perfect prediction line

residuals <- testcars$Price - predict(lss_lm, newdata = testcars)

# Plot histogram
hist(residuals,
     main = "Histogram of lss log power (Actual - Predicted Price)",
     xlab = "Residual (Price - Predicted)",
     col = "skyblue",
     border = "white",
     breaks = 10)

lss_lm_1 <- lm(Price ~ Brand + Fuel_Type + Mileage + Transmission + log(Power) + Kilometers_Driven + Seats, data = traincars)
lss_lm_1

Call:
lm(formula = Price ~ Brand + Fuel_Type + Mileage + Transmission + 
    log(Power) + Kilometers_Driven + Seats, data = traincars)

Coefficients:
       (Intercept)            BrandBMW           BrandFord          BrandHonda        BrandHyundai       BrandMahindra         BrandMaruti       BrandMercedes  
        -3.350e+06           4.111e+04          -8.563e+05          -7.524e+05          -9.958e+05          -1.067e+06          -8.543e+05           4.853e+04  
         BrandTata         BrandToyota     BrandVolkswagen     Fuel_TypePetrol             Mileage  TransmissionManual          log(Power)   Kilometers_Driven  
        -7.661e+05          -7.167e+05          -7.145e+05          -1.429e+05          -2.984e+04          -1.602e+05           1.241e+06          -1.074e+01  
             Seats  
         5.325e+04  

predicted_prices <- predict(lss_lm_1, newdata = testcars)
plot(testcars$Price, predicted_prices,
     xlab = "Actual Price",
     ylab = "Predicted Price",
     main = "Actual vs Predicted Prices",
     pch = 19, col = "blue")
abline(0, 1, col = "red", lwd = 2)  # perfect prediction line

residuals <- testcars$Price - predict(lss_lm_1, newdata = testcars)

# Plot histogram
hist(residuals,
     main = "Histogram of lss log power (Actual - Predicted Price)",
     xlab = "Residual (Price - Predicted)",
     col = "skyblue",
     border = "white",
     breaks = 10)

par(mfrow=c(2,2))
#Choose variables in lasso to perform a regular linear regression.
lss_lm = lm(formula = Price ~ Brand + Fuel_Type + Mileage + Transmission + Power + Kilometers_Driven + Seats, data = traincars)
plot(lss_lm)